404 XV NORMED ALGEBRAS AND SPECTRAL THEORY 6. Let E be a separable Hilbert space and T a continuous operator on E Let R and L be the positive hermitian operators which are the square roots (15.1112 of r*T and TT\ respectively. We write R - abs(D, and call it the absolute value of T. Then Z, = 'abs(T*). ___ ----- (a) Show that Ker(D = KerGR) and that L(E) = T(E). There exists a unique isometry Vof R(E) onto T(E) such that T= VR. If we extend Vby continuity to R(E), and then to an operator Ue *(£) by putting U(x) = 0 on the orthogonal supplement of then we have also T=UR (polar decomposition of T). Show that R = CW= U*UR = RU*U, L = URU", T= LU*. fb) For T to be invertible it is necessary and sufficient that K = abs(T) and 1 = abs(r*) are invertible. (To prove necessity, consider the spectra of R and L. To prove sufficiency, use the closed graph theorem.) ...... (c) JVis normal if and only if abs(AO = absGV*), and if this condition is satisfied there exists a unitary operator W such that N = W- abs(AO. 7 A compact operator T on a separable Hilbert space E is said1 to.be nuclear if denoting by (A/the full sequence of eigenvalues of abs(T) (Section 11.5, Problem 8), we have a) Use polar decomposition (Problem 6) to show that the product SA of two Hilbert-Schmidt operators is nuclear. Conversely, if T is nuclear, then abs r)>" is a self-adjoint Hilbert-Schmidt operator, and T is the product of two Hilbert- Schmidt operators. Consequently T* is also nuclear. If A is any continuous operator on E, then AT and TA are nuclear. . (b) If A and B are two Hilbert-Schmidt operators and if <>„) is a Hilbert basis of E, then the series£ (A3 • *n I *„) and^ (BA • en \ *„) are absolutely convergent, and their sums are equal."(Write B • e, =£ (B • e. \ e»)em.) Consequently, for every unitary operator t/and every nuclear operator T, we have £ (U~1TU• en | ea) =£ (T-en\ ea). n " Deduce that, for a nuclear operator T, the sum £ (T- <?„ | O >s independent of the Hilbert basis (*„) chosen. This sum is called the trace of 7" and is denoted by Tr(T). If A, B are two Hilbert-Schmidt operators, then Tr(^-B) = Tr(BA) = (A \ B*). (c) If T is nuclear, show that Tr(abs(r» = supfeK3"' a» 16»>l) where the supremum is taken over all pairs of Hilbert bases (an), (ba) of E Se the polar decomposition of E). If we put \\T\\t = Tr(abs(D), then the set ^(E) of nuclear operators on E is a vector space on which ||r||, is a norm, such that II T"ll <I II Til d) I?(T.Hs a sequence of nuclear (resp. Hilbert-Schmidt) operators on E which converges weakly (Section 12.15, Problem 9) to an operator T, and which is such that the sequence of norms (||r,|U) (resp. (||r,||a)) is bounded, then Tis a nuclear (resp, Hilbert-Schmidt) operator, (e) Show that ^?i(E) is a Banach space with respect to the norm || / ||,.quently u — a + ib is unitary, and a =